Optimal quantum noise cancellation with an entangled witness channel

We present the digital signal processing of a mutually entangled, two-mode squeezed state using Wiener filtering to maximize the reduction of quantum noise of a single mode. By conditioning this mode, the signal, with its directly detected entangled pair, the witness, we show quantum noise cancellation of 2 dB below that of the signal vacuum level. We present the frequency-dependent digital recovery of squeezed states with Wiener filtering. This filtering is particularly relevant for gravitational wave detectors which will seek to use frequency-dependent squeezed states to improve their reach to the observable universe. We demonstrate the recovery of squeezed states in a configuration that replicates one which would provide optimum sensitivity improvement in a gravitational wave detector under the effects of radiation pressure noise. More generally, this technique may find application in other quantum-limited high-precision experiments such as those using optomechanical cavities.

Figure 1

Simplified schematic of the experiment. An OPO pumped with a second harmonic field at $2{\omega }_0+\mathrm{\Delta }$ produces the entangled sidebands of the signal (red) and witness (blue) fields within the linewidth of the separate resonances of the OPO. Inset (a) shows the frequency spacing of these fields under the OPO resonances. The entangled fields are directed to the test cavity where the signal field is resonant and the witness field is detuned. Inset (b) shows this as a frequency spacing diagram with the resonances of the test cavity shown in black. The witness field picks up a phase rotation from the phase response of the test cavity upon reflection. A Faraday isolator separates the reflected beams, sending them to a triangular mode cleaner designed such that the signal is transmitted and witness reflected, separating the two fields. The fields are detected with separate homodyne detectors for which local oscillators are generated symmetrically around $\omega +\frac{\mathrm{\Delta }}{2}$ by strongly modulating a tap off from the main laser frequency with an EOM. A comprehensive description of the experimental design is provided in the Supplemental Material of [41].

Figure 2

(a) Noise spectra normalized to the shot noise of a single homodyne detector. The test cavity in this case is far detuned for both signal and witness fields. The purple trace shows the shot noise seen by the other homodyne. The black trace shows the dark noise of the measurement system. The raw signal (orange) and witness (blue) channels show the variance of each entangled field. These fields are are both approximately 12 dB above the shot-noise level, their slight difference due to the difference in losses on the two squeezing paths caused by the mode cleaner tuning. The two sub-shot-noise traces show a linear filter of $-0.95$ (red) and the Wiener filter (green). (b) Magnitude and phase of the Wiener filter.

Figure 3

(a) Noise spectra normalized to the shot noise of a single homodyne detector. Both signal and witness fields are detuned by half the linewidth of the test cavity. The signal and witness fields undergo the same phase change resulting in a rotation of the squeezed quadrature around the test cavity linewidth. The two linear filters are shown together with the results of a fitting routine [43]. In dark blue and light blue we have the $-0.85$ filter spectrum and its fitted model; in red and orange we have the $+0.85$ filter spectrum and its model. (b) Applied Wiener filter magnitude and phase shown together with a model that uses the parameters produced by the fitting routine.

Figure 4

(a) Noise spectra normalized to the shot noise of a single homodyne detector. The test cavity has here been tuned such that the signal field is resonant while the witness is detuned. The two linear filters are shown together with the results of a fitting routine [43]. In dark blue and light blue we have the $-0.9$ filter spectrum and its fitted model; in red and orange we have the $+0.9$ filter spectrum and its model. (b) Magnitude and phase of the Wiener filter shown together with its model taking the parameters from the fitting routine.